An equation concerning with the subdifferential of convex functionals defined in real Banach spaces and the metric projections to level sets is shown. The equation is compared with the resolvents of general monotone o...An equation concerning with the subdifferential of convex functionals defined in real Banach spaces and the metric projections to level sets is shown. The equation is compared with the resolvents of general monotone operators, and makes the geometric properties of differential equations expressed by subdifferentials clear. Hence, it can be expected to be useful in obtaining the steepest descents defined by the convex functionals in Banach spaces. Also, it gives a similar result to the Lagrange multiplier method under certain conditions.展开更多
In the presem paper, some important characteristics of Fenchel-, Frechet-,Hademard-, and Gateaux-Subdifferentials are showed up, and properties of functions, especially. convexity of functions, are described by subdif...In the presem paper, some important characteristics of Fenchel-, Frechet-,Hademard-, and Gateaux-Subdifferentials are showed up, and properties of functions, especially. convexity of functions, are described by subdifferentials.展开更多
In this paper, the existence theorem of the cone weak subdifferential of set valued mapping in locally convex topological vector space is proved. Received March 30,1998. 1991 MR Subject Classification: 4...In this paper, the existence theorem of the cone weak subdifferential of set valued mapping in locally convex topological vector space is proved. Received March 30,1998. 1991 MR Subject Classification: 47H17,90C29.展开更多
For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smalle...For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smallest enclosing ball problem.Some properties of the maximal time function are proven,including the convexity,the lower semicontinuity,and the exact characterizations of its subdifferential formulas.展开更多
This paper first shows that for any p∈(1,2)there exists a continuously differentiable function f on l^(p)(and L^(p))such that the proximal subdifferential of f is empty everywhere,and hence it is not suitable to deve...This paper first shows that for any p∈(1,2)there exists a continuously differentiable function f on l^(p)(and L^(p))such that the proximal subdifferential of f is empty everywhere,and hence it is not suitable to develop theory on proximal subdifferential in the classical Banach spaces l^(P)and L^(P) with p∈(1,2).On the other hand,this paper establishes variational analysis based on the proximal subdifferential in the framework of smooth Banach spaces of power type 2,which conclude all Hilbert spaces and all the classical spaces l^(P)and L^(P)with p∈(2,+∞).In particular,in such a smooth space,we provide the proximal subdifferential rules for sum functions,product functions,composite functions and supremum functions,which extend the basic results on the proximal subdifferential established in the framework of Hilbert spaces.Some of our main results are new even in the Hilbert space case.As applications,we provide KKT-like conditions for nonsmooth optimization problems in terms of proximal subdifferential.展开更多
Abstract Recently a, monotone generalized directional derixrative has been introduced for Lipschitz functions. This concept has been applied to represent and optimize nonsmooth functions. The second a.pplication resul...Abstract Recently a, monotone generalized directional derixrative has been introduced for Lipschitz functions. This concept has been applied to represent and optimize nonsmooth functions. The second a.pplication result,ed relevant for parallel computing, by allowing to define minimization algorithms with high degree of inherent parallelism. The paper presents first the theoretical background, namely the notions of monotone generalized directional derivative and monotone generalized subdifferential. Then it defines the tools for the procedures, that is a necessary optimality condition and a steel>est descent direction. Therefore the minimization algorithms are outlined. Successively the used architectures and the performed numerical expertence are described, by listing and commenting the t.ested functions and the obtained results.展开更多
文摘An equation concerning with the subdifferential of convex functionals defined in real Banach spaces and the metric projections to level sets is shown. The equation is compared with the resolvents of general monotone operators, and makes the geometric properties of differential equations expressed by subdifferentials clear. Hence, it can be expected to be useful in obtaining the steepest descents defined by the convex functionals in Banach spaces. Also, it gives a similar result to the Lagrange multiplier method under certain conditions.
文摘In the presem paper, some important characteristics of Fenchel-, Frechet-,Hademard-, and Gateaux-Subdifferentials are showed up, and properties of functions, especially. convexity of functions, are described by subdifferentials.
文摘In this paper, the existence theorem of the cone weak subdifferential of set valued mapping in locally convex topological vector space is proved. Received March 30,1998. 1991 MR Subject Classification: 47H17,90C29.
基金supported by the National Natural Science Foundation of China(11201324)the Fok Ying Tuny Education Foundation(141114)the Sichuan Technology Program(2022ZYD0011,2022NFSC1852).
文摘For a general normed vector space,a special optimal value function called a maximal time function is considered.This covers the farthest distance function as a special case,and has a close relationship with the smallest enclosing ball problem.Some properties of the maximal time function are proven,including the convexity,the lower semicontinuity,and the exact characterizations of its subdifferential formulas.
基金Supported by the National Natural Science Foundation of P.R.China(Grant No.12171419)。
文摘This paper first shows that for any p∈(1,2)there exists a continuously differentiable function f on l^(p)(and L^(p))such that the proximal subdifferential of f is empty everywhere,and hence it is not suitable to develop theory on proximal subdifferential in the classical Banach spaces l^(P)and L^(P) with p∈(1,2).On the other hand,this paper establishes variational analysis based on the proximal subdifferential in the framework of smooth Banach spaces of power type 2,which conclude all Hilbert spaces and all the classical spaces l^(P)and L^(P)with p∈(2,+∞).In particular,in such a smooth space,we provide the proximal subdifferential rules for sum functions,product functions,composite functions and supremum functions,which extend the basic results on the proximal subdifferential established in the framework of Hilbert spaces.Some of our main results are new even in the Hilbert space case.As applications,we provide KKT-like conditions for nonsmooth optimization problems in terms of proximal subdifferential.
文摘Abstract Recently a, monotone generalized directional derixrative has been introduced for Lipschitz functions. This concept has been applied to represent and optimize nonsmooth functions. The second a.pplication result,ed relevant for parallel computing, by allowing to define minimization algorithms with high degree of inherent parallelism. The paper presents first the theoretical background, namely the notions of monotone generalized directional derivative and monotone generalized subdifferential. Then it defines the tools for the procedures, that is a necessary optimality condition and a steel>est descent direction. Therefore the minimization algorithms are outlined. Successively the used architectures and the performed numerical expertence are described, by listing and commenting the t.ested functions and the obtained results.
